basic_engineering_mathematics0

16
Reduction of non-linear laws to linear form
16.1 Determination of law
Frequently, the relationship between two variables, say x and y,
is not a linear one, i.e. when x is plotted against y a curve results.
In such cases the non-linear equation may be modified to the
linear form, y = mx + c, so that the constants, and thus the law
relating the variables can be determined. This technique is called
‘determination of law’.
Some examples of the reduction of equations to linear form
include:
(i) y = ax 2 + b compares with Y = mX + c, where m = a, c = b
and X = x 2 .
Hence y is plotted vertically against x 2 horizontally to
produce a straight line graph of gradient ‘a’ and y-axis
intercept ‘b’
(ii) y = a x + b
y is plotted vertically against 1 horizontally to produce a
x
straight line graph of gradient ‘a’ and y-axis intercept ‘b’
(iii) y = ax 2 + bx
Dividing both sides by x gives y = ax + b.
x
If y is plotted against x a curve results and it is not possible
to determine the values of constants a and b from the curve.
Comparing y = ax 2 + b with Y = mX + c shows that y is to be
plotted vertically against x 2 horizontally. A table of values is
drawn up as shown below.
x 1 2 3 4 5
x 2 1 4 9 16 25
y 9.8 15.2 24.2 36.5 53.0
A graph of y against x 2 is shown in Fig. 16.1, with the best straight
line drawn through the points. Since a straight line graph results,
the law is verified.
y
53
50
40
30
A
Comparing with Y = mX + c shows that y is plotted vertically
against x horizontally to produce a straight line graph
x
of gradient ‘a’ and y axis intercept ‘b’
x
20
17
10
8
0 5 10 15 20 25 x 2
C
B
Problem 1. Experimental values of x and y, shown
below, are believed to be related by the law
y = ax 2 + b. By plotting a suitable graph verify this
law and determine approximate values of a and b.
x 1 2 3 4 5
y 9.8 15.2 24.2 36.5 53.0
Fig. 16.1
From the graph, gradient a = AB 53 − 17
=
BC 25 − 5 = 36
20 = 1.8